The applet is designed to impart a geometric understanding of why this is true. It graphs the image in the complex plane, through the entered polynomial, of the circle of radius r. For small r, this is approximately a small circle around the constant term. For very large r, this is approximately a large circle that wraps n times around the origin, where n is the degree of the polynomial. For topological reasons, at some r value in between, the image must pass through the origin. When it does, a root is found.

This applet lets you vary the radius and search out these roots.

The real and imaginary parts of the polynomial must be entered separately in the function entering panels at the bottom of the applet in this version. There are instructions for how to enter other functions into these applets, but probably you should just try to enter things in and experiment -- always use * for multiplication, and ^ for powers, and make reasonable guesses about function names, and you may not need the instructions.

Also, when you click to go to the radius entering panel, click again after you get there. For reason unbeknownst to me, the canvas on which the radius and such is reported erases itself after being drawn in. But a second click brings it back. The second click makes the exact same graphics calls, so this shouldn't happen. In any case, a second click cures it. If you know how to solve this -- the source is available on-line -- please let me know.

The applets take a while to load -- they are built out of a lot of building blocks of code, and do lots of things. But once downloaded, at least if you're going to do the ptoject, there is a lot to do with them.